International Journal of Engineering Mathematics

Volume 2016, Article ID 6390367, 18 pages

http://dx.doi.org/10.1155/2016/6390367

## A New Accurate and Efficient Iterative Numerical Method for Solving the Scalar and Vector Nonlinear Equations: Approach Based on Geometric Considerations

Aix-Marseille Université, IFSTTAR, LBA UMR T24, 13016 Marseille, France

Received 31 March 2016; Accepted 12 June 2016

Academic Editor: Josè A. Tenereiro Machado

Copyright © 2016 Grégory Antoni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with a new numerical iterative method for finding the approximate solutions associated with both scalar and vector nonlinear equations. The iterative method proposed here is an extended version of the numerical procedure originally developed in previous works. The present study proposes to show that this new root-finding algorithm combined with a stationary-type iterative method (e.g., Gauss-Seidel or Jacobi) is able to provide a longer accurate solution than classical Newton-Raphson method. A numerical analysis of the developed iterative method is addressed and discussed on some specific equations and systems.

#### 1. Introduction

Solving both nonlinear equations and systems is a situation very often encountered in various fields of formal or physical sciences. For instance, solid mechanics is a branch of physics where the resolution of problems governed by nonlinear equations and systems occurs frequently [1–10]. In most cases, Newton method (also known as Newton-Raphson algorithm) is most commonly used for approximating the solutions of scalar and vector nonlinear equations [11–13]. But, over the years, several other numerical methods have been developed for providing iteratively the approximate solutions associated with nonlinear equations and/or systems [14–25]. Some of them present the advantage of having both high accuracy and strong efficiency using a numerical procedure based on an enhanced Newton-Raphson algorithm [26]. In this study, we propose to improve the iterative procedure developed in previous works [27, 28] for finding numerically the solution of both scalar and vector nonlinear equations. This study is decomposed as follows: (i) in Section 2, a new numerical geometry-based root-finding algorithm coupled with a stationary-type iterative method (such as Jacobi or Gauss-Seidel) is presented with the aim of solving any system of nonlinear equations [29, 30]; (ii) in Section 3, the numerical predictive abilities associated with the proposed iterative method are tested on some examples and also compared with other algorithms [31, 32].

#### 2. New Iterative Numerical Method for Solving the Scalar and Vector Nonlinear Equations Based on a Geometric Approach

##### 2.1. Problem Statement

We consider vector-valued function , which is continuous and infinitely differentiable (i.e., ), checking the following equation:where denotes the vector-valued variable (with ), is th component associated with vector (with ), is the transpose operator associated with the variable , and denotes the class of infinitely differentiable functions in domain . It should be mentioned that: (i) the nonlinear function has a unique solution on domain which is an open subset of , that is, such as ; (ii) the case of scalar equation () with only one variable () is obtained when , that is, .

Equation (1) can also rewritten as system of -scalar nonlinear equations, that is,where denotes th component associated with vector-valued function (see (1)), that is, th nonlinear equation of system . It should be noted that: (i) in the case of (with ), the nonlinear system (2) has a unique solution set such as ; (ii) in the case of , nonlinear system (2) is transformed in scalar nonlinear equation which has a unique solution such as ; (iii) (1) and (2) are mathematically equivalent, that is, .

With the aim of numerically solving system (2), we adopt a Root-Finding Algorithm (RFA) coupled with a Stationary Iterative Procedure (SIP) such as Jacobi or Gauss-Seidel [26, 30]. The use of any SIP allows to reduce the considered nonlinear system to a successive set of nonlinear equations with only one variable and therefore it can be solved with a RFA [30]. In the present study, we propose an extended version of RFA already developed in [27, 28] and combined with a Jacobi or Gauss-Seidel type iterative procedure for dealing any system of nonlinear equations.

##### 2.2. Stationary Iterative Procedures (SIPs) with Root-Finding Algorithms (RFAs)

###### 2.2.1. Jacobi and Gauss-Seidel Iterative Procedures

In order to solve a system of nonlinear equations, any RFA can be used if it is combined with a SIP (i.e., Jacobi or Gauss-Seidel) [26, 29, 30]. A Jacobi or Gauss-Seidel type procedure applied to nonlinear system (1) can be described as follows:with(i)in the case of Jacobi procedure:(ii)in the case of Gauss-Seidel procedure:where (resp., ) denotes th (resp., th) iteration associated with the variable (), is the set of kept constant variables, and represents one set of variables .

##### 2.3. Used Root-Finding Algorithm (RFA)

In previous works [27, 28], a root-finding algorithm (RFA) has been developed for approximating the solutions of scalar nonlinear equations. The new RFA presented here is an extended version to that previously developed taking into account some geometric considerations. In this paper, we propose to use a RFA coupled with Jacobi and Gauss-Seidel type procedures for iteratively solving nonlinear system . Hence, we adopt a new RFA for finding approximate solution (when is fixed and with the known set ) associated with each nonlinear equation of system (see Section 2.1). For each nonlinear equation , parametrized by the set of variables , depending only on one variable , we introduce the exact and inexact local curvature associated with the curve representing the nonlinear equation in question.

The used RFA is based on the following main steps (see [27] for more details):(i)In the first step, we consider the iterative tangent and normal straight lines associated with nonlinear function at point (see Figure 1): where (resp., ) denotes the value (resp., first-order derivative) of function at point , is the set of known variables and are two functionals associated with .(ii)In the second step, we introduce the iterative exact and inexact local curvature associated with the curve representing nonlinear function at point (see Figure 1): where denotes the absolute-value function associated with the variable , is the exact () or inexact () radius of the osculating at point , (with ) is functional associated with , and is the second-order derivative of function at point . It should be noted that: (i) we consider the exact radius associated with the true osculating circle at point (see [33]); (ii) in line with [27, 28], we consider an inexact radius associated with the osculating circle at point , that is, (see (7)).(iii)In the third step, we define the iterative center associated with the exact and inexact osculating circle at point , that is, (see Figure 1) By taking (7) and (8), the iterative centers are (with ) where is the iterative centers associated with the exact and inexact osculating circle (with ) associated with each curve representing nonlinear function at point , are two functionals associated with and is sign function (i.e., when , when , and when ).(iv)In the fourth step, we introduce the iterative point such as , that is, where is a functional associated with .(v)In the fifth step, we define the iterative straight line passing through two iterative points and , that is, (with ) where is the set of known variables.(vi)In the sixth step, we introduce the iterative straight line passing through the point and the iterative perpendicular straight line such as (with ) where is a functional associated with .(vii)In the last step, we define the iterative point which is the solution of the following relation (with ) with where is a functional associated with . In line with (10), (14) can be rewritten as